Chapter 1 Introduction Imagine the geometry of four-dimensional space done with a view to learning about the living conditions of spirits. Does this mean that it is not mathematics? Ludwig Wittgenstein, Remarks on the Foundations of Mathematics In China, from about the ﬁrst century C.E. through the seventeenth century, anonymous and most likely illiterate adepts practiced an arcane art termed fangcheng (often translated into English as “matrices” or “rectangular arrays”). This art entailed procedures for manipulating counting rods arranged on a counting board,1 which enabled practitioners to produce answers to seemingly insoluble riddles. The art seems to have been closely aligned with other mathematical arts (suanfa ), including various forms of calculation, numerology, and possibly divination. Though we know virtually nothing about these practitioners, records of their practices have been preserved. These practices were occasionally recorded by aspiring literati and incorporated in texts they compiled on the mathematical arts, which were then presented to the imperial court, along with prefaces claiming that these mathematical arts were essential to ordering the empire. In their prefaces and commentaries, these literati sometimes assigned credit to mysterious recluses, apparently to support the claimed exclusivity and authenticity of their compilation; yet at other times the literati compilers also denounced fangcheng practitioners for employing overly arcane techniques, apparently in an attempt to reassert their own higher status and authority. Over this period of perhaps sixteen centuries, bibliographies of imperial libraries recorded the titles of hundreds of treatises on the mathematical arts. Many of these are still extant, and many include fangcheng problems. The earliest of these texts were collected in imperial libraries, and later reprinted under imperial auspices. From these efforts arose a textual tradition in which literati collected the early canonical texts, offered commentaries for them, and ultimately sought to recover their original meanings. 1 For an explanation of the counting board, see chapter 2 of this book. 1 2 1 Introduction Fangcheng, an art that apparently found no practical application beyond solving implausible riddles, is remarkable for several reasons: it is essentially equivalent to the solution of systems of n equations in n unknowns in modern linear algebra; the earliest recorded fangcheng procedure is in many ways quite similar to what we now call Gaussian elimination. These procedures were transmitted to Japan, and there is a reasonable possibility that they were also transmitted to Europe, serving perhaps as precursors of modern matrices, Gaussian elimination, and determinants. In any case, there was very little work on linear algebra in Greece or Europe prior to Gottfried Leibniz’s studies of elimination and determinants, beginning in 1678. As is well known, Leibniz was a Sinophile interested in the translations of such Chinese texts as were available to him, and in particular the Classic of Changes (Yi jing ). Overview of This Book This is the ﬁrst book-length study in any language of linear algebra in imperial China; it is also the ﬁrst book-length study of linear algebra as it existed before 1678, the date Leibniz began his studies. My central purpose is to reconstruct fangcheng practices using extant textual sources. My focus is therefore on presenting an in-depth study of the mathematics behind these early texts: extant Chinese treatises on the mathematical arts are, after all, translations into narrative form of calculations originally performed on a counting board. The Argument The central argument of this book is that the essential feature in the solution of fangcheng problems is the visualization of the problem in two dimensions as an array of numbers on a counting board and the “cross multiplication” of entries , which led to general solutions of systems of linear equations not found in Greek or early European mathematics. There are, I argue, two distinct types of “cross multiplication” found in extant Chinese mathematical treatises, which correspond to the two methods used to solve problems in modern linear algebra today : (1) wei cheng , the “cross multiplication” of individual entries, as found in the “excess and deﬁcit” (ying bu zu ) procedure in the Nine Chapters on the Mathematical Arts;2 and (2) bian cheng , the “cross multiplication” by individual elements of an entire column, as found in the fangcheng procedure (in modern terms, “row operations,” since the columns in fangcheng arrays correspond to rows in matrices in modern linear algebra). The latter type, the “cross multiplication” of an entire column, corresponds to Gaussian elimination, 2 For a discussion of the Nine Chapters on the Mathematical Arts, see chapter 3 of this book. Overview...

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